Maximal partial packings of Zn2 with perfect codes

Thomas Westerbäck*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)

Abstract

A maximal partial Hamming packing of Zn2 is a family S of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in S . The number of translates of Hamming codes in S is the packing number, and a partial Hamming packing is strictly partial if the family S does not constitute a partition of S. A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of Zn2, n = 2 m -1, satisfies m+1 ≤ p ≤ n-1 . It is also proved that for any n equal to 2 m -1, m ≥ 4 , there exist maximal strictly partial Hamming packings of Zn2 with packing numbers n-10,n-9,n-8,...,n-1. This implies that the upper bound is tight for any n = 2 m -1, m ≥ 4 . All packing numbers for maximal strictly partial Hamming packings of Zn2 , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.

Original languageEnglish
Pages (from-to)335-355
Number of pages21
JournalDESIGNS CODES AND CRYPTOGRAPHY
Volume42
Issue number3
DOIs
Publication statusPublished - Mar 2007
MoE publication typeA1 Journal article-refereed

Keywords

  • Perfect codes

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